Conditionality Properties for the Bivariate Logarithmic Distribution with an Application to Goodness of Fit

  • A. W. Kemp
Part of the NATO Advanced study Institutes Series book series (ASIC, volume 79)


The paper gives new modes of genesis for the multivariate and homogeneous bivariate logarithmic distributions. Marginality and conditionality properties are derived (including, e.g., X|Y > X, X|Y - X = n), and are applied to the problem of grouping frequencies for a chi-square goodness-of-fit test for the homogeneous bivariate logarithmic distribution. The proposed procedure is objective and impartial with respect to rows and columns.

Key Words

Bivariate logarithmic series distribution multivariate logarithmic series distribution modes of genesis chi-squared goodness-of-fit objective grouping homogeneous distributions 


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  1. Bates, G.E., Neyman, J. (1952). Contributions to the theory of accident proneness. University of California Publications in Statistics, 1, 215–275.MathSciNetGoogle Scholar
  2. Boswell, M.T., Patil, G.P., (1971). Chance mechanisms generating the logarithmic series destribution used in the analysis of number of species and individuals. In Statistical Ecology, Vol. 1 G.P. Patil et al The Pennsylvania state University Press, University Park. Pages 99–130Google Scholar
  3. Chatfield, C., Ehrenberg, A.S.C., Goodhardt, G. J. (1966). Progress on a simplified model of stationary purchasing behaviour. Journal of the Royal Statistical Society, Series A, 129, 317–367.CrossRefGoogle Scholar
  4. Edwards, C.B., Gurland, J. (1961). A class of distributions applicable to accidents. Journal of the American Statistical Association, 56, 503–517.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Fisher, R.A., Corbet, A.S., Williams, C.B. (1943). The relation between the number of species and the number of individuals in a random sample from an animal population. Journal of Animal Ecology, 12, 42–58.CrossRefGoogle Scholar
  6. Guldberg, A. (1934). On discontinuous frequency functions of two variables. Skandinavisk Aktuarietidscrift, 17, 89–117.zbMATHGoogle Scholar
  7. Johnson, N.L., Kotz, S. (1969). Discrete Distributions. Houghton Mifflin, Boston.zbMATHGoogle Scholar
  8. Katti, S.K. (1967). Infinite divisibility of integer-valued random variables. Annals of Mathematical Statistics, 38, 1306–1308.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Kemp, A.W. (1978). Cluster size probabilities for generalized Poisson distributions. Communications in Statistics, Series A, 7, 1433–1438.MathSciNetCrossRefGoogle Scholar
  10. Kemp, A.W. (1979). Convolutions involving binomial pseudo- variables. Sankhyā, Series A, 41, 232–243.MathSciNetzbMATHGoogle Scholar
  11. Kemp, A.W. (1981). Frugal methods of generating bivariate discrete random variables. In Statistical Distributions in Scientific Work, C. Taillie, G. P. Patil, B. Baldessari, eds. Reidel, Dordrecht-Holland. Vol. 4, 321–329.Google Scholar
  12. Kemp, A.W. (1981). Efficient generation of the logarithmic distribution. Applied Statistics (in press).Google Scholar
  13. Kemp, A.W., Kemp, C.D. (1968). On a distribution associated with certain stochastic processes. Journal of the Royal Statistical Society, Series B, 30, 160–163.MathSciNetGoogle Scholar
  14. Kemp, A.W., Kemp, C.D. (1969). Branching and clustering models associated with the ‘lost-games’ distribution. Journal of Applied Probability, 6, 700–703.MathSciNetzbMATHCrossRefGoogle Scholar
  15. Kendall, D.G. (1948). On some modes of population growth leading to R.A. Fisher’s logarithmic series distribution. Biometrika, 35, 6–15.MathSciNetzbMATHGoogle Scholar
  16. Khatri, C.G. (1959). On certain properties of power series distributions. Biometrika, 46, 486–490.MathSciNetzbMATHGoogle Scholar
  17. McCloskey, J.W. (1965). A model for the distribution of individuals by species in an environment. Technical Report No. 7, Department of Statistics, Michigan State University.Google Scholar
  18. Nelson, W.C., David, H.A. (1967). The logarithmic distribution: a review. Virginia Journal of Science, 18, 95–102.Google Scholar
  19. Neyman, J. (1965). Certain chance mechanisms involving discrete distributions. In Classical and Contagious Discrete Distributions, G. P. Patil, ed. Pergamon Press, London. Pages 4–14.Google Scholar
  20. Patil, G.P., Bildikar, S. (1967). Multivariate logarithmic series distribution as a probability model in population and community ecology and some of its statistical properties. Journal of the American Statistical Association, 62, 655–674.MathSciNetzbMATHCrossRefGoogle Scholar
  21. Patil, G.P., Joshi, S.W. (1968). A Dictionary and Bibliography of Discrete Distributions. Oliver and Boyd, Edinburgh.zbMATHGoogle Scholar
  22. Philippou, N., Roussas, G.G. (1974). A note on the multi-variate logarithmic series distribution. Communications in Statistics, 3, 469–472.MathSciNetCrossRefGoogle Scholar
  23. Quenouille, M.H. (1949). A relation between the logarithmic, Poisson and negative binomial series. Biometrics, 5, 162–164.MathSciNetCrossRefGoogle Scholar
  24. Sibuya, M., Yoshimura, I., Shimizu, R. (1964). Negative multinomial distribution. Annals of the Institute of Statistical Mathematics Tokyo, 26, 409–426.MathSciNetCrossRefGoogle Scholar
  25. Steutel, F.W. (1968). A class of infinitely divisible mixtures. Annals of Mathematical Statistics, 39, 1153–1157.MathSciNetzbMATHCrossRefGoogle Scholar
  26. Subrahmanian, K. (1966). A test for “intrinsic correlation” in the theory of accident proneness. Journal of the Royal Statistical Society, Series B, 28, 180–189.Google Scholar

Copyright information

© D. Reidel Publishing Company 1981

Authors and Affiliations

  • A. W. Kemp
    • 1
  1. 1.School of Mathematical SciencesUniversity of BradfordBradfordEngland

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